Tight Bounds for Distributed Graph Computations

Motivated by the need to understand the algorithmic foundations of distributed large-scale graph computations, we study some fundamental graph problems in a message-passing model for distributed computing where $k \geq 2$ machines jointly perform computations on graphs with $n$ nodes (typically, $n \gg k$). We present (almost) tight bounds for the round complexity of two fundamental graph problems, namely triangle enumeration and PageRank computation. Our tight lower bounds, a main contribution of the paper, are established through an information-theoretic approach that relates the round complexity to the minimal amount of information required by machines for solving a problem. Our approach, as demonstrated in the case of triangle enumeration, can yield stronger round lower bounds as well as message-time tradeoffs compared to approaches that use communication complexity techniques. We then present algorithms that (almost) match the lower bounds; these algorithms exhibit a round complexity which scales superlinearly in $k$, improving significantly over previous results. Specifically, we show the following results: 1. Triangle enumeration: We show a lower bound of $\tilde{\Omega}(m/k^{5/3})$ rounds, where $m$ is the number of edges of the graph. ($\tilde \Omega$ hides a $1/\text{polylog}(n)$ factor; $\tilde O$ hides a $\text{polylog}(n)$ factor and an additive $\text{polylog}(n)$ term.) This result implies the first non-trivial lower bound of $\tilde\Omega(n^{1/3})$ rounds for the congested clique model. We also present a distributed algorithm that enumerates all the triangles of a graph in $\tilde{O}(m/k^{5/3})$ rounds. 2. PageRank: We show a lower bound of $\tilde{\Omega}(n/k^2)$ rounds and an $\tilde{O}(n/k^2)$ rounds algorithm.